I have experienced this "stiffness" and, as other writers have said, there may be many reasons for this. Stress may be one reason. If we are serious about being the best we can, our shoulders can tighten and rise up giving a stiff appearance. To dance latin or rhythm dances well you need to be able to control all of your body parts precisely giving the illusion that this is easy to do. Of course, it is not. Core strength, and a considerable amount, is essential but not the only requirement. A stable frame is also essential but if overly firm it will appear stiff. Leads that are too extreme or abrupt may appear as stiffness. Styling with the arms must be fluid yet stop sharply for those picture moments. Jerkyness can pull your shoulders up and produce robot-like motion.
Good dancers are a total package of strength, balance and absolute control over every movement. It takes time and a lot of hard work to get there. I'm convinced that you never stop learning and growing.
Good luck!